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Chapter 5: Problem 28

Change each radical to simplest radical form. \(\sqrt{112}\)

Short Answer

Expert verified
The simplest radical form of \(\sqrt{112}\) is \(4\sqrt{7}\).

Step by step solution

01

Factor the Number Inside the Radical

To simplify \( \sqrt{112} \), we first need to find the prime factorization of 112. Factor 112 as: 112 = 2 脳 56 Further factoring 56, we get: 56 = 2 脳 28 Continue factoring 28 as: 28 = 2 脳 14 Finally, factor 14 as: 14 = 2 脳 7.So, 112 = 2 脳 2 脳 2 脳 2 脳 7 = 2^4 脳 7.
02

Pair the Common Factors

Next, look for pairs of the same prime factors within the factorization. From 112, we have four 2's, which can form two pairs: \(2^4\) can be written as \((2^2) \times (2^2)\).
03

Simplify the Radical

Use the rule \( \sqrt{a^2} = a \) to bring each pair out of the square root as a single number:\( \sqrt{2^4 \times 7} = \sqrt{(2^2) \times (2^2) \times 7}\)Which simplifies to:\( 2 \times 2 \times \sqrt{7} = 4 \sqrt{7} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization
Prime factorization is a process of breaking down a number into its basic building blocks, which are prime numbers. These are numbers that can only be divided by 1 and themselves without leaving a remainder. When simplifying radicals like \( \sqrt{112} \), the first step is to express the number inside the radical as a product of prime factors.
For example, let's consider the number 112:
  • Begin by dividing 112 by the smallest prime number, which is 2: \( 112 \div 2 = 56 \)
  • Continue factoring 56, again by 2: \( 56 \div 2 = 28 \)
  • Next, factor 28: \( 28 \div 2 = 14 \)
  • Finally, factor 14: \( 14 \div 2 = 7 \)
Now, 7 is a prime number, so it cannot be factored further. So, the prime factorization of 112 is \( 2 \times 2 \times 2 \times 2 \times 7 \) or \( 2^4 \times 7 \). By breaking numbers down into prime factors, we set the stage for further simplification.
Pairing Factors
When you have broken down a number into its prime factors, the next step toward simplifying a radical is to look for pairs of the same factors. This is useful because pairs allow us to simplify the square root more easily.
Let's take the factorization of 112 from earlier: \( 2^4 \times 7 \). We need to identify pairs:
  • The term \( 2^4 \) means we have four 2s; two 2s make a pair.
  • So, \( 2^4 \) can be rearranged into two pairs: \((2 \times 2) \times (2 \times 2)\).
Pairing factors is a crucial step because each pair can be taken out of the square root. In this case, for every pair of 2 inside the radical, one 2 can be brought out, simplifying our calculation significantly.
Square Root Simplification
Once you have paired similar factors, simplifying the square root becomes straightforward. The rule to remember is: \( \sqrt{a^2} = a \). This means, when you have a pair of numbers within a square root, you can bring them out as a single number.

Take our previous example: \( \sqrt{(2^2) \times (2^2) \times 7} \).
  • Each pair of \( 2^2 \) inside the square root becomes a single 2 outside of it. So, we bring two 2s out.
  • Thus, \( \sqrt{2^4 \times 7} \) simplifies to \( 2 \times 2 \times \sqrt{7} \).
  • Multiplying the numbers outside the square root gives \( 4 \sqrt{7} \).
Simplifying square roots in this way helps make calculations easier and gives the radical in its simplest form. Remember, only pairs of factors can be simplified, and any unpaired factors remain inside the radical.

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Most popular questions from this chapter

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(0.0214\)

Use your calculator to estimate each of the following to the nearest one- thousandth. (a) \(7^{\frac{4}{3}}\) (b) \(10^{\frac{4}{5}}\) (c) \(12^{\frac{3}{5}}\) (d) \(19^{\frac{2}{5}}\) (e) \(7^{\frac{3}{4}}\) (f) \(10^{\frac{5}{4}}\)

Definition \(5.7\) states that $$ b^{\frac{m}{n}}=\sqrt[n]{b^{m}}=(\sqrt[n]{b})^{m} $$ Use your calculator to verify each of the following. (a) \(\sqrt[3]{27^{2}}=(\sqrt[3]{27})^{2}\) (b) \(\sqrt[3]{8^{5}}=(\sqrt[3]{8})^{5}\) (c) \(\sqrt[4]{16^{3}}=(\sqrt[4]{16})^{3}\) (d) \(\sqrt[3]{16^{2}}=(\sqrt[3]{16})^{2}\) (e) \(\sqrt[5]{9^{4}}=(\sqrt[5]{9})^{4}\) (f) \(\sqrt[3]{12^{4}}=(\sqrt[3]{12})^{4}\)

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(0.347\)

Explain the importance of scientific notation.
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